Research article Special Issues

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

The stochastic extinction and stability conditions for nonlinear malaria epidemics

Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3042, Statesboro, Georgia, 30460, USA

Special Issues: Mathematical Modeling of Mosquito-Borne Diseases

## Abstract    Full Text(HTML)    Figure/Table    Related pages

The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number $R^{*}_{0}$, the expected survival probability of the plasmodium $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, and other threshold values are calculated, where $\mu_{v}$ and $\mu$ are the natural death rates of mosquitoes and humans, respectively, and $T_{1}$ and $T_{2}$ are the incubation periods of the plasmodium inside the mosquitoes and humans, respectively. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local Martingale and Lyapunov functional techniques are applied to establish the strong persistence, and asymptotic stability in the mean of the malaria-free steady population. %The extinction of malaria depends on $R^{*}_{0}$, and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$. Moreover, for either $R^{*}_{0}<1$, or $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$, whenever $R^{*}_{0}\geq 1$, respectively, extinction of malaria occurs. Furthermore, the robustness of these threshold conditions to the intensity of noise from the disease transmission rate is exhibited. Numerical simulation results are presented.
Figure/Table
Supplementary
Article Metrics

Citation: Divine Wanduku. The stochastic extinction and stability conditions for nonlinear malaria epidemics. Mathematical Biosciences and Engineering, 2019, 16(5): 3771-3806. doi: 10.3934/mbe.2019187

References

• 1. World malaria report 2017, Licence: CC BY-NC-SA 3.0, World Health Organization, Geneva, 2017.
• 2. D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, Int. J. Biomath., 11 (2018), 1850085–1850130.
• 3. Dengue control, Report of World Health Organization, 2019. Available from: http://www.who. int/denguecontrol/human/en/.
• 4. Malaria, Report of Global Health, Division of Parasitic Diseases and Malaria, 2019. Available from: https://www.cdc.gov/malaria/about/disease.html.
• 5. J. M. Crutcher and S. L. Hoffman, Malaria, Chapter 83-malaria, Medical Microbiology, 4th edi-tion, Galveston (TX), University of Texas Medical Branch at Galveston, 1996.
• 6. D. L. Doolan, C. Dobano and J. K. Baird, Acquired Immunity to Malaria, Clin. Microbiol. Rev.,22 (2009), 13–36.
• 7. L. Hviid, Naturally acquired immunity to Plasmodium falciparum malaria, Acta Trop., 95 (2005), 270–275.
• 8. E. Avila and V. Buonomo, Analysis of a mosquito-borne disease transmission model with vector stages and nonlinear forces of infection, Ric. Mat., 64 (2015), 377–390.
• 9. Z. Bai and Y. Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Anal. Real World Appl., 13 (2012), 1060–1068.
• 10. M. De la Sena, S. Alonso-Quesadaa and A. Ibeasb, On the stability of an SEIR epidemic model withdistributedtime-delayandageneralclassoffeedbackvaccinationrules, Appl.Math.Comput.,270 (2015), 953–976.
• 11. M. De la Sen, S. Alonso-Quesada and A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953–976.
• 12. N. H. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Appl. Math. Lett., 64 (2017), 223–230.
• 13. Z. Jiang, W. Ma and J. Wei, Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Math. Comput. Simulation, 122 (2016), 35–54.
• 14. Q. Liu, D. Jiang, N. Shi, et al., Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence, Commun. Nonlinear Sci. Numer. Simul., 40 (2016), 89–99.
• 15. Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 428 (2015), 140–153.
• 16. J. P. Mateus and C. M. Silva, Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Anal. Real World Appl., 34 (2017), 379–402.
• 17. J. P. Mateus and C. M. Silva, A non-autonomous SEIRS model with general incidence rate, Appl. Math. Comput., 247 (2014), 169–189.
• 18. L. Pang, S. Ruan, S. Liu, et al., Transmission dynamics and optimal control of measles epidemics,Appl. Math. Comput., 256 (2015), 131–147.
• 19. S. Syafruddin and M. Salmi Md. Noorani, Lyapunov function of SIR and SEIR model for trans-mission of dengue fever disease, Int. J. Simul. Process Model., 8 (2013), 2–3.
• 20. D. Wanduku, Complete Global Analysis of a Two-Scale Network SIRS Epidemic Dynamic Model with Distributed Delay and Random Perturbation,J. Appl. Math. Comput., 294 (2017), 49–76.
• 21. D. Wanduku and G. S. Ladde, Fundamental Properties of a Two-scale Network stochastic human epidemic Dynamic model, Neural Parallel Sci. Comput., 19 (2011), 229–270.
• 22. R. Ross, The Prevention of Malaria, John Murray, London, 1911.
• 23. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
• 24. N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272–1296.
• 25. M. Y. Hyun, Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector). Rev. Saude Publica, 34 (2000), 223–231.
• 26. G. Macdonald, The analysis of infection rates in diseases in which superinfection occurs. Trop. Dis. Bull., 47 (1950), 907–915.
• 27. G. A. Ngwa and W. Shu, A mathematical model for endemic malaria with variable human and mosquito population, Math. computer. model., 32 (2000), 747–763.
• 28. G. A. Ngwa, A. M. Niger and A. B. Gumel, Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector, Appl. Math. Comput.,217 (2010), 3286–3313.
• 29. C. N. Ngonghala , G. A. Ngwa and M. I. Teboh-Ewungkem, Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission. Math. Biosci., 240 (2012), 45–62.
• 30. M. I. Teboh-Ewungkem and T. Yuster, A within-vector mathematical model of plasmodium fal-ciparum and implications of incomplete fertilization on optimal gametocyte sex ratio, J. Theory Biol., 264 (2010), 273–286.
• 31. R. Reiner, M. Geary, P. Atkinson, et al., Seasonality of Plasmodium falciparum transmission: a systematic review, Malar. J, 14 (2015), 343.
• 32. A. Teklehaimanot and P. Mejia, Malaria and poverty, Ann. N. Y. Acad. Sci., 1136 (2008), 32–37.
• 33. D. Wanduku and G. S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear Anal. Real World Appl., 13 (2012), 794–816.
• 34. E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delay influ-enced by stochastic perturbations, Math. Comput. Simulat., 45 (1998), 269–277.
• 35. E. J. Allen, L. J. S. Allen, A. Arciniega, et al., Construction of equivalent stochastic differential equation models, Stoch. Anal. Appl. , 26 (2008), 274–297.
• 36. Y. Cai, J. jiao, Z. Gui, et al., Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210–226.
• 37. M. Krstic, The effect of stochastic perturbation on a nonlinear delay malaria epidemic model,Math. Comput. Simulat., 82 (2011), 558–569.
• 38. P. V. V. Le, P. Kumar and M. O. Ruiz, Stochastic lattice-based modelling of malaria dynamics,Malar. J., 17 (2018), 250.
• 39. D. Wanduku, A comparative stochastic and deterministic study of a class of epidemic dynamic models for malaria: exploring the impacts of noise on eradication and persistence of disease, preprint arXiv:1809.03897.
• 40. Y. Cai, Y. Kang and W. Wang, a stochastic SIRS epidemic model with nonlinear incidence, Appl. Math. Comput., 305 (2017), 221–240.
• 41. Y. Cai, Y. kang, M. Banerjee, et al., A stochastic epidemic model incorporating media coverage,Commun. math sci., 14 (2016), 893–910.
• 42. A. Gray, D. Greenhalgh, L. Hu, et al., A Stochastic Differential Equation SIS Epidemic Model,SIAM J. Appl. Math., 71 (2011), 876–902.
• 43. A. lahrouz and L. Omari, extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83 (2013), 960–968
• 44. Y. zhang, K. Fan, S. Gao, et al., A remark on stationary distribution of a stochastic SIR epidemic model with double saturated rates, Appl. Math. Lett., 76 (2018), 46–52.
• 45. Y. Zhou, W. Zhang, S. Yuan, et al., Persistence And Extinction in Stochastic Sirs Models With General Nonlinear Incidence Rate, Electron. J. Differ. Eq., 2014 (2014), 1–17.
• 46. K. L. Cooke, Stability analysis for a vector disease model. Rocky Mountain J. Math., 9 (1979), 31–42.
• 47. C. McCluskey, Global Stability of an SIR epidemic model with delay and general nonlinear inci-dence, Math. Biosci. Eng., 4 (2010), 837–850.
• 48. Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931–947.
• 49. D. Wanduku, Modeling Highly Random Dynamical Infectious Systems, in Applied Mathemati-cal Analysis: Theory, Methods, and Applications (eds. Dutta H., Peters J.), Studies in Systems, Decision and Control, 177 (2020), Springer, Cham.
• 50. D. Wanduku and G. S. Ladde, The global analysis of a stochastic two-scale Network Human Epidemic Dynamic Model With Varying Immunity Period, J. Appl. Math. Phys., 5 (2017), 1150–1173.
• 51. D. Wanduku and G. S. Ladde, Global Stability of Two-Scale Network Human Epidemic Dynamic Model, Neural Parallel Sci. Comput., 19 (2011), 65–90.
• 52. M. Zhien and C. Guirong, Persistence and extinction of a population in a polluted environment,Math Biosci., 101 (1990), 75–97.
• 53. X. Zhang, D. Jiang, T. Hayat, et al., Dynamics of stochastic SIS model with doubles epidemic diseases driven by levy jumps, Phys. A, 47 (2017), 767–777.
• 54. X. Mao, Stochastic differential equations and applications, 2nd ed, Horwood Publishing Ltd., 2008.
• 55. D.Wanduku, SomestochasticstabilitypropertiesofanonlinearfamilyofSEIRSepidemicmodels, in press, 2019.
• 56. S. Ruan, D. Xiao and J. C. Beier, On the delayed rossmacdonald model for malaria transmission.Bull. Math. Biol., 70 (2008), 1098–1114.
• 57. R. E. Howes, K. E. Battle, K. N. Mendis, et al., Global Epidemiology of Plasmodium vivax, Am. J. Trop. Med. Hyg., 95 (2016), 15–34.
• 58. Aedes aegypti in Brazil, Report of OXITEC, 2019. Available from: https://www.oxitec.com/friendly-mosquitoes/brazil/.
• 59. Piracicaba, SP, Report of Atlas of Human Development in Brazil, 2019. Available from: http: //www.atlasbrasil.org.br/2013/en/perfil_m/piracicaba_sp/.
• 60. The World Factbook, Report of Central Intelligence Agency, 2019. Available from: https://www.cia.gov/library/publications/the-world-factbook/geos/br.html.
• 61. World Malaria Report 2011, Report of World Health Organization, 2011. Available from: https: //www.who.int/malaria/publications/atoz/9789241564403/en/.
• 62. M. T Bretscher, N. Maire, I. Felger, et al. , Asymptomatic Plasmodium falciparum infections may not be shortened by acquired immunity, Malar. J., 14 (2015), 294.
• 63. S. M. Moghadas and A. B. Gumel, Global Stability of a two-stage epidemic model with general-ized nonlinear incidence, Math. Comput. Simulat., 60 (2002), 107–118.
• 64. K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math Biol. 35 (1996), 240–260.